Advanced information feedback in intelligent traffic systems Wen-Xu Wang,* Bing-Hong Wang, Wen-Chen Zheng, Chuan-Yang Yin, and Tao Zhou Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei Anhui, 230026, People’s Republic of China 共Received 30 January 2005; revised manuscript received 7 September 2005; published 5 December 2005兲 The optimal information feedback is very important to many socioeconomic systems like stock market and traffic systems aiming to make full use of resources. As to traffic flow, a reasonable real-time information feedback can improve the urban traffic condition by providing route guidance. In this paper, the influence of a feedback strategy named congestion coefficient feedback strategy is introduced, based on a two-route scenario in which dynamic information can be generated and displayed on the board to guide road users to make a choice. Simulation results adopting this optimal information feedback strategy have demonstrated high efficiency in controlling spatial distribution of traffic patterns compared with the other two information feedback strategies, i.e., travel time and mean velocity. DOI: 10.1103/PhysRevE.72.066702

PACS number共s兲: 89.40.⫺a, 64.60.Ak, 02.60.Cb

II. THE MODEL AND FEEDBACK STRATEGIES

I. INTRODUCTION

Traffic flow, a kind of multibody system consisting of interacting vehicles, shows various complex behaviors. Therefore, in the past few decades, problems of traffic systems have triggered great interest of a community of physicists 关1–3兴 and many theories such as kinetic theory 关4–10兴, car-following theory 关11兴, and particle-hopping theory 关12,13兴 have been introduced with the aim of alleviating the traffic congestion and enhance the capacity of the existing infrastructure. Although dynamics of traffic flow with realtime traffic information have been extensively investigated 关14–19兴, finding a more efficient feedback strategy is an overall task. Recently, some real-time feedback strategies have been put forward, such as travel time feedback strategy 共TTFS兲 关14,20兴 and mean velocity feedback strategy 共MVFS兲 关14,21兴. It has been proved that MVFS is more efficient than that of TTFS which brings a lag effect to make it impossible to provide the road users with the real situation of each route 关21兴. However, MVFS is still not the best one due to the fact that the random brake mechanism of the Nagel-Schreckenberg 共NS兲 model 关22兴 brings fragile stability of velocity and some other reasons which will be discussed delicately in this paper. In order to provide road users with better guidance, a strategy named congestion coefficient feedback strategy 共CCFS兲 is presented. We report the simulation results adopting three different feedback strategies in a two-route scenario with each single route following the NS the mechanism. The outline of this paper is as follows: in the next section the NS model and two-route scenario are briefly introduced, together with three feedback strategies of TTFS, MVFS, and CCFS all depicted in more detail. In Sec. III some simulation results will be presented and discussed based on the comparison of three different feedback strategies. The last section will make some conclusions.

*Electronic address: [email protected] 1539-3755/2005/72共6兲/066702共6兲/$23.00

A. NS mechanism

The Nagel-Schreckenberg 共NS兲 model is so far the most popular and simplest cellular automaton model in analyzing the traffic flow 关1–3,22,23兴, where the one-dimension CA with periodic boundary conditions is used to investigate highway and urban traffic. This model can reproduce the basic features of real traffic like stop-and-go wave, phantom jams, and the phase transition on a fundamental diagram. In this section, the NS mechanism will be briefly introduced as a base of analysis. The road is subdivided into cells with a length of ⌬x = 7.5 m. Let N be the total number of vehicles on a single route of length L, then the vehicle density is = N / L. gn共t兲 is defined to be the number of empty sites in front of the nth vehicle at time t, and vn共t兲 to be the speed of the nth vehicle, i.e., the number of sites that the nth vehicle moves during the time step t. In the NS model, the maximum speed is fixed to be vmax = M. In the present paper, we set M = 3 for simplicity. The NS mechanism can be decomposed to the following four rules 共parallel dynamics兲: Rule 1. Acceleration: vi ← min共vi + 1 , M兲; Rule 2. Deceleration: vi⬘ ← min共vi , gi兲; Rule 3. Random brake: with a certain brake probability P do vi⬙ ← max共vi⬘ − 1 , 0兲; and Rule 4. Movement: xi ← xi + vi⬙. The fundamental diagram characterizes the basic properties of the NS model which has two regimes called “freeflow” phase and “jammed” phase. The critical density, basically depending on the random brake probability p, divides the fundamental diagram to these two phases. B. Two-route scenario

Wahle et al. 关20兴 first investigated the two-route model in which road users choose one of the two routes according to the real-time information feedback. In the two-route scenario, it is supposed that there are two routes A and B of the same length L. At every time step, a new vehicle is generated

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at the entrance of two routes and will choose one route. If a vehicle enters one of two routes, the motion of it will follow the dynamics of the NS model. As a remark, if a new vehicle is not able to enter the desired route, it will be deleted. The vehicle will be removed after it reaches the end point. Additionally, two types of vehicles are introduced: dynamic and static vehicles. If a driver is a so-called dynamic one, he will make a choice on the basis of the information feedback 关20兴, while a static one just enters a route at random ignoring any advice. The density of dynamic and static travelers are Sdyn and 1 − Sdyn, respectively. The simulations are performed by the following steps: first, set the routes and board empty; then, after the vehicles enter the routes, according to three different feedback strategies, information will be generated, transmitted, and displayed on the board at every time step. Then the dynamic road users will choose the route with better condition according to the dynamic information at the entrance of two routes. C. Related definitions

The roads conditions can be characterized by flux of two routes, and flux is defined as follows: N F = Vmean = Vmean , L

共1兲

where Vmean represents the mean velocity of all the vehicles on one of the roads, N denotes the vehicle number on each road, and L is the length of two routes. Then we describe three different feedback strategies, respectively. TTFS: At the beginning, both routes are empty and the information of travel time on the board is set to be the same. Each driver will record the time when he enters one of the routes. Once a vehicle leaves the two-route system, it will transmit its travel time on the board and at that time a new dynamic driver will choose the road with shorter time. MVFS: Every time step, each vehicle on the routes transmits its velocity to the traffic control center which will deal with the information and display the mean velocity of vehicles on each route on the board. Road users at the entrance will choose one road with larger mean velocity. CCFS: Every time step, each vehicle transmits its signal to satellite, then the navigation system 共GPS兲 will handle that information and calculate the position of each vehicle which will be transmitted to the traffic control center. The work of the traffic control center is to compute the congestion coefficient of each road and display it on the board. Then drivers can choose one road guided by the information on the board. The congestion coefficient is defined as m

C = 兺 nwi .

共2兲

i=1

Here, ni stands for vehicle number of the ith congestion cluster in which cars are close to each other without a gap between any two of them. Every cluster is evaluated a weight w, here w = 2, see Fig. 1 共w = 1 shows no point for it just indicates the vehicle number and one can check out that w ⬎ 2 leads to the similar results with w = 2兲. The reason for

FIG. 1. Illustration of two routes with different congestion coefficient C. Each route has three clusters. According to Eq. 共2兲, Ca = 14, Cb = 41. Apparently, condition of route a is better than that of route b, which can be accurately reflected by C.

adding weight to each cluster can be explained by the fact that travel time of the last vehicle of the cluster from the entrance to the destination is obviously affected by the size of cluster. Imagine that with the increasing of cluster size, travel time of the last vehicle will be more and more longer and the correlation between cluster size and travel time of the last vehicle is nonlinear. For simplicity, an exponent w is added to the size of each cluster to be consistent with the nonlinear relationship. In the following section, performance by using three different feedback strategies will be shown and discussed in more detail. III. SIMULATION RESULTS

All simulation results shown here are obtained by 30 000 iterations excluding the initial 5000 time steps. In contrast with MVFT and CCFT, the flux of two routes adopting TTFS shows oscillation obviously due to the information lag effect. This lag effect can be understood as that the travel time reported by a driver at the end of two routes only represents the road condition in front of him, and perhaps the vehicles behind him have got into the jammed state. Unfortunately, this information will induce more vehicles to choose his route until a vehicle from the jammed cluster leaves the system. This effect apparently does harm to the system. Compared to MVFS, the performance adopting CCFS is remarkably improved, not only on the value but also the stability of the flux. Therefore as to the flux of the two-route system, CCFS is the best one 共see Fig. 2兲. In Fig. 3, vehicle number versus time step shows almost the same tendency as Fig. 2, the routes’ accommodating capacity is greatly enhanced with an increase in vehicle number from 270 to 330, so perhaps the high flux of two routes with CCFS are mainly due to the increase of vehicle number. The lag effect by TTFS also leads to severe amplitude oscillation in figures of travel time 共Fig. 4兲 and vehicle speed 共Fig. 5兲, but the performance adopting CCFS does not show much difference compared with mean velocity feedback strategy, but even behaves slightly bad in stability than MVFS. There are two reasons to explain why MVFS is not the optimal strategy. We have mentioned that the NS model has a random brake scenario which causes the fragile stability of velocity, so MVFS cannot completely reflect the real condition of routes. The other reason is that flux consists of two parts, mean velocity and vehicle density, but MVFS only grasps one part and lacks the other part of flux. Maybe someone will ask why we do not use the flux feedback strategy? Although adopting flux feedback strategy can indeed make full use of existing infrastructure of two routes, the cost of

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FIG. 2. 共Color online兲 共a兲 Flux of each route with travel time and congestion coefficient feedback strategies. 共b兲 Flux of each route with mean velocity feedback strategy. The parameters are L = 2000, p = 0.25, and sdyn = 0.5.

this strategy makes it not worthwhile to do it. As to CCFS, it avoids the shortcoming of insufficient information of road condition and large amount of expense and only needs the information of each vehicle’s position. So CCFS is the most appropriate strategy at present. Figure 6 shows that the average flux fluctuates feebly with a persisting increase of dynamic travelers by using the new strategy. As to the routes’ processing capacity, the new strategy is proved to be the most proper one because the flux is always the largest at each Sdyn value and keeps the two routes’ flux in balance. In this case the average flux of the two routes almost shows no change. So this will avoid any influence on using road conditions due to variation of unpredictable proportion of Sdyn. In succession, we will discuss how the length of routes affects the average flux of two routes adopting three different

strategies. One can see from Fig. 7 that in the range of the length less than 1000 cells, the average flux adopting MVFS and TTFS decreases severely with increasing L. This property indicates that compared to the other two strategies, average flux with C feedback almost does not depend on the changing of route length, and C feedback is the optimal strategy among them from this aspect. Generally speaking, the road users who do not follow the information feedback may not choose route A or B completely at random, they may have their own preference to one of the routes, or perhaps some of them like to follow suit or not follow suit, so the real road users without information feedback are not the same as the static users in the model. However, if all road users choose routes completely at random, the routes can be utilized approximately in balance, which is why the average flux of the two routes adopting

FIG. 3. 共Color online兲 共a兲 Vehicle number of each route with travel time and congestion coefficient feedback strategies. 共b兲 Vehicle number of each route with mean velocity feedback strategy. The parameters are set the same as in Fig. 2.

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FIG. 4. 共Color online兲 Travel time of each route adopting 共a兲 congestion coefficient feedback strategy, 共b兲 travel time feedback strategy, and 共c兲 mean velocity feedback strategy. The dark line and red dotted line represent route 1 and route 2, respectively. The parameters are set the same as in Fig. 2.

three kinds of information feedback cannot overweight the flux of two routes chosen completely at random. However, our feedback strategy can make the roads be fully used as that selected completely at random, meanwhile it makes existing infrastructure used more efficiently than that without information feedback. Furthermore, we investigate how the traffic properties adopting CCFS are influenced by the information feedback delay. From the perspective of technology, to realize CCFS the delay of information feedback should be considered. In Fig. 8, we depict the average flux of two routes as a function

of feedback period. We find that the average flux decreases slightly when the period is short. While for the very long period, the average flux reaches a lower limit the same as in the case of adopting TTFS. Due to the feedback delay, the information displayed on the board cannot reflect the realtime route condition, which leads to the overbalance in the utilization of the two routes during the feedback period. Therefore one route gets more crowded than the other one and the velocity on this congested route decreases, thus the average flux will decrease. Figure 9 shows the vehicle number of two routes affected by the feedback period. It is found

FIG. 5. 共Color online兲 Average speed of each route by using 共a兲 congestion coefficient feedback strategy, 共b兲 mean velocity feedback strategy, and 共c兲 travel time feedback strategy. The dark line and red dotted line represent route 1 and route 2, respectively. The parameters are set the same as in Fig. 2.

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FIG. 6. 共Color online兲 Average flux by performing different strategy vs Sdyn; L is fixed to be 2000.

that the vehicle number of two route reduces with increasing feedback period. For a short period, vehicle number does not exhibit considerable change compared to the case of no feedback delay, which indicates that the two-route system still performs satisfyingly in the case of short information feedback delay. While for long feedback delay, for example, when period= 1000, the oscillation of the tworoute vehicle number emerges and the system behaves in an undesirable way. As to the TTFS, the oscillation behaviors are also found, which are caused by the lag effect. Therefore we can conclude that the large delay of real-time information will lead to the oscillation of two-route IV. CONCLUSION utilization. We obtain the simulation results of applying three different feedback strategies, i.e., TTFS, MVFS, and CCFS on a

two-route scenario all with respect to travel time, speed, number of cars, average flux, average flux versus Sdyn, and length. The results indicates that the CCFS strategy has more advantages than the two former ones. The highlight of this paper is that it brings forward a new and better quantity namely congestion coefficient to radically describe road conditions. In contrast with the two old strategies, the CCFS strategy can bring a significant improvement to the road conditions, including increasing vehicle number and flux, reducing oscillation, and that average flux does not reduce with increase of Sdyn and route lengths. The numerical simulations demonstrate that the congestion coefficient is meaningful and a basic quantity for describing the road condition.

FIG. 7. 共Color online兲 Average flux vs route length L by using three different strategies. The parameter is Sdyn = 0.9.

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FIG. 8. Average flux vs information feedback period by adopting CCFS. The parameters are L = 2000, p = 0.25, and sdyn = 0.5.

FIG. 9. Vehicle number of two routes as a function of time step by adopting CCFS for a different information feedback period. The parameters are the same as Fig. 8.

Due to the rapid development of modern scientific technology, it is not difficult to realize CCFS. If only a navigation system 共GPS兲 is installed in each vehicle, thus the position information of vehicles will be known, then the CCFS strategy can come true and also it will cost no more than MVFS. Taking into account the reasonable cost and more accurate description of road conditions, we think that this strategy shall be applicable.

This work has been partially supported by the State Key Development Program of Basic Research 共973 Project兲 of China, the National Natural Science Foundation of China 共under Grant Nos. 70271070, 1047216, and 10532060兲, and the Specialized Research Fund for the Doctoral Program of Higher Education 共SRFDP Grant No. 20020358009兲.

关1兴 D. Chowdhury, L. Santen, and A. Schadschneider, Phys. Rep. 329, 199 共2000兲. 关2兴 D. Helbing, Rev. Mod. Phys. 73, 1067 共2001兲. 关3兴 T. Nagatani, Rep. Prog. Phys. 65, 1331 共2002兲. 关4兴 I. Prigogine and F. C. Andrews, Oper. Res. 8, 789 共1960兲. 关5兴 S. L. Paveri-Fontana, Transp. Res. 9, 225 共1975兲. 关6兴 H. Lehmann, Phys. Rev. E 54, 6058 共1996兲. 关7兴 C. Wagner, C. Hoffmann, R. Sollacher, J. Wagenhuber, and B. Schürmann, Phys. Rev. E 54, 5073 共1996兲. 关8兴 D. Helbing, Phys. Rev. E 53, 2366 共1996兲. 关9兴 D. Helbing, Phys. Rev. E 57, 6176 共1997兲. 关10兴 D. Helbing and M. Treiber, Phys. Rev. Lett. 81, 3042 共1998兲. 关11兴 R. W. Rothery, in Traffic Flow Theory, edited by N. Gartner, C. J. Messner, and A. J. Rathi, Transportation Research Board Special Report, Vol. 165 共Transportation Research Board, Washington, D.C., 1992兲, Chap. 4. 关12兴 K. Nagel and M. Schreckenberg, J. Phys. I 2, 2221 共1992兲. 关13兴 O. Biham, A. A. Middleton, and D. Levine, Phys. Rev. A 46, R6124 共1992兲.

关14兴 Y. Yokoya, Phys. Rev. E 69, 016121 共2004兲. 关15兴 T. L. Friesz, J. Luque, R. L. Tobin, and B.-W. Wie, Oper. Res. 37, 893 共1989兲. 关16兴 M. Ben-Akiva, A. de Palma, and I. Kaysi, Transp. Res., Part A 25A, 251 共1991兲. 关17兴 H. S. Mahmassani and R. Jayakrishnan, Transp. Res., Part A 25A, 293 共1991兲. 关18兴 R. Arnott, A. de Palma, and R. Lindsey, Transp. Res., Part A 25A, 309 共1991兲. 关19兴 P. Kachroo and K. Özbay, Transp. Res. Rec. 1556, 137 共1996兲. 关20兴 J. Wahle, A. L. C. Bazzan, F. Klügl, and M. Schreckenberg, Physica A 287, 669 共2000兲. 关21兴 K. Lee, P.-M. Hui, B.-H. Wang, and N. F. Johnson, J. Phys. Soc. Jpn. 70, 3507 共2001兲. 关22兴 K. Nagel and M. Schreckenberg, J. Phys. I 2, 2221 共1992兲. 关23兴 B.-H. Wang, D. Mao, and P.-M. Hui, Proceedings of The Second International Symposium on Complexity Science, Shanghai, August 6–7, 2002, p. 204.

ACKNOWLEDGMENTS

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